| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + i·11-s + 14-s + 16-s − i·17-s − i·19-s − i·22-s + i·23-s − 28-s + 29-s + i·31-s − 32-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + i·11-s + 14-s + 16-s − i·17-s − i·19-s − i·22-s + i·23-s − 28-s + 29-s + i·31-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7409926425 - 0.3342791886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7409926425 - 0.3342791886i\) |
| \(L(1)\) |
\(\approx\) |
\(0.6350932820 - 0.03953947770i\) |
| \(L(1)\) |
\(\approx\) |
\(0.6350932820 - 0.03953947770i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.724540300145548818940512596047, −26.17800360364742310576996111770, −25.1553677836099954419301507920, −24.34447711133443631067462362632, −23.21643872228145970280859850681, −21.98643554284597016950098656587, −21.03710759569358263903774825982, −19.92839528823422263040347679348, −19.09161859397945879591519097488, −18.48033622435763919229346151407, −17.10630359726862079430948915133, −16.44212322860531234463876148498, −15.5838369075532692713451657390, −14.3669187153450335594305733579, −12.957275639553225677496789264685, −11.986419918008030613624870199238, −10.72475369559553584949605306534, −9.982562706913970289011792352187, −8.8162932689438771092144832272, −7.97444095694929437697622541166, −6.55713846334184624471620331571, −5.88234653243166917438940807912, −3.74445377029960313116819160753, −2.57641043747673282277784130647, −0.92278513549557170414136021272,
0.505338892164766350982565530222, 2.18095088018424318407059749145, 3.37775562834642620040205462931, 5.206905308777858151301084089476, 6.73092713109467595217874905639, 7.27508801558363414153626135864, 8.792900273152612623767092402978, 9.6112619883035092822296095996, 10.4516077482395146274725759036, 11.74458282727361491723237088584, 12.6184201115701756666009593132, 13.95394708536171787038323985063, 15.52081035176229472289109730206, 15.871546040309653036997970343489, 17.19940104435949661238795777947, 17.88416567253355949052117213164, 19.024743327205456589361084170360, 19.78601231106169870514028058023, 20.58906500366481002165347987484, 21.79345361487512176551155133136, 22.90169605141579588762672926864, 23.92493563388358623520364452468, 25.26012893102429822086077463557, 25.60160503927075562184985089416, 26.64592941514705716743911789755
